Step |
Hyp |
Ref |
Expression |
1 |
|
dnizeq0.t |
⊢ 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) ) |
2 |
|
dnizeq0.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
3 |
2
|
zred |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 |
1
|
dnival |
⊢ ( 𝐴 ∈ ℝ → ( 𝑇 ‘ 𝐴 ) = ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) ) |
5 |
3 4
|
syl |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝐴 ) = ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) ) |
6 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ℝ ) |
8 |
2 7
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ℤ ∧ ( 1 / 2 ) ∈ ℝ ) ) |
9 |
|
flzadd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 1 / 2 ) ∈ ℝ ) → ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) = ( 𝐴 + ( ⌊ ‘ ( 1 / 2 ) ) ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) = ( 𝐴 + ( ⌊ ‘ ( 1 / 2 ) ) ) ) |
11 |
6
|
rexri |
⊢ ( 1 / 2 ) ∈ ℝ* |
12 |
|
0re |
⊢ 0 ∈ ℝ |
13 |
|
halfgt0 |
⊢ 0 < ( 1 / 2 ) |
14 |
12 6 13
|
ltleii |
⊢ 0 ≤ ( 1 / 2 ) |
15 |
|
halflt1 |
⊢ ( 1 / 2 ) < 1 |
16 |
11 14 15
|
3pm3.2i |
⊢ ( ( 1 / 2 ) ∈ ℝ* ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) < 1 ) |
17 |
|
0xr |
⊢ 0 ∈ ℝ* |
18 |
|
1xr |
⊢ 1 ∈ ℝ* |
19 |
17 18
|
pm3.2i |
⊢ ( 0 ∈ ℝ* ∧ 1 ∈ ℝ* ) |
20 |
|
elico1 |
⊢ ( ( 0 ∈ ℝ* ∧ 1 ∈ ℝ* ) → ( ( 1 / 2 ) ∈ ( 0 [,) 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ* ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) < 1 ) ) ) |
21 |
19 20
|
ax-mp |
⊢ ( ( 1 / 2 ) ∈ ( 0 [,) 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ* ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) < 1 ) ) |
22 |
16 21
|
mpbir |
⊢ ( 1 / 2 ) ∈ ( 0 [,) 1 ) |
23 |
22
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ( 0 [,) 1 ) ) |
24 |
|
ico01fl0 |
⊢ ( ( 1 / 2 ) ∈ ( 0 [,) 1 ) → ( ⌊ ‘ ( 1 / 2 ) ) = 0 ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → ( ⌊ ‘ ( 1 / 2 ) ) = 0 ) |
26 |
25
|
oveq2d |
⊢ ( 𝜑 → ( 𝐴 + ( ⌊ ‘ ( 1 / 2 ) ) ) = ( 𝐴 + 0 ) ) |
27 |
3
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
28 |
27
|
addid1d |
⊢ ( 𝜑 → ( 𝐴 + 0 ) = 𝐴 ) |
29 |
26 28
|
eqtrd |
⊢ ( 𝜑 → ( 𝐴 + ( ⌊ ‘ ( 1 / 2 ) ) ) = 𝐴 ) |
30 |
10 29
|
eqtrd |
⊢ ( 𝜑 → ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) = 𝐴 ) |
31 |
30
|
oveq1d |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) = ( 𝐴 − 𝐴 ) ) |
32 |
27
|
subidd |
⊢ ( 𝜑 → ( 𝐴 − 𝐴 ) = 0 ) |
33 |
31 32
|
eqtrd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) = 0 ) |
34 |
33
|
fveq2d |
⊢ ( 𝜑 → ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) = ( abs ‘ 0 ) ) |
35 |
|
abs0 |
⊢ ( abs ‘ 0 ) = 0 |
36 |
35
|
a1i |
⊢ ( 𝜑 → ( abs ‘ 0 ) = 0 ) |
37 |
34 36
|
eqtrd |
⊢ ( 𝜑 → ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) = 0 ) |
38 |
5 37
|
eqtrd |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝐴 ) = 0 ) |